Force‐stepping integrators in Lagrangian mechanics
Force‐stepping integrators in Lagrangian mechanics
复制标题
拉格朗日力学中的力步进积分器
DOI：
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发表时间：
2010
期刊：
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通讯作者：
Magdalena Ortiz
中科院分区：
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作者：
M. González;B. Schmidt;Magdalena Ortiz
我们提出了一种拉格朗日力学的积分方案，称为力步进方案，它是辛的，节能的，时间可逆的，并且具有自动选择时间步长的收敛性。该方案还近似地保留了与系统对称性相关的所有动量映射。动量映射的精确守恒还可以借助拉格朗日约化来实现。力步进格式是通过在简单网格或规则三角剖分上用分段仿射近似代替势能得到的。通过采用尺寸递减的三角剖分，可以生成一个近似的能量序列。所得到的近似拉格朗日量的轨迹可以明确地表征，并由分段抛物运动或自由落体组成。选定的数值试验证明了力步进的良好长期性能，其自动时间步长选择特性，以及处理约束（包括接触问题）的便利性。版权所有©2010 John Wiley & Sons， Ltd
We formulate an integration scheme for Lagrangian mechanics, referred to as the force‐stepping scheme, which is symplectic, energy conserving, time‐reversible, and convergent with automatic selection of the time‐step size. The scheme also conserves approximately all the momentum maps associated with the symmetries of the system. The exact conservation of momentum maps may additionally be achieved by recourse to the Lagrangian reduction. The force‐stepping scheme is obtained by replacing the potential energy by a piecewise affine approximation over a simplicial grid or regular triangulation. By taking triangulations of diminishing size, an approximating sequence of energies is generated. The trajectories of the resulting approximate Lagrangians can be characterized explicitly and consist of piecewise parabolic motion, or free fall. Selected numerical tests demonstrate the excellent long‐term behavior of force‐stepping, its automatic time‐step selection property, and the ease with which it deals with constraints, including contact problems. Copyright © 2010 John Wiley & Sons, Ltd.